Note that in both [1]} and [2]} there was a mistake – the prefactor for \(A(\varepsilon )\)  in the definition of \(\psi ^\varepsilon _w(x)\)  should be \(\frac{N(N-1)}{2}\)  (as above) whereas in [1]} and [2]} it was written as \(\frac{(N-1)(N+2)}{2}\) . Let us briefly explain where this error came from (in reference to the proof of [2]}). Comparing equation (3.8) with (4.25) we find that in (4.25) the term \(\Delta ^{\varepsilon }({x}_{\ell })\)  should actually be \(\Delta ^{\varepsilon }({x}_{\ell +1})\)  (here \(\ell +1=N\) ). This error resulted in neglecting \(\ell +1\)  extra factors of \(e^{A(\varepsilon )}\)  which jives with the difference between the incorrect and correct powers \(\frac{(N-1)(N+2)}{2} - \frac{N(N-1)}{2} = N\) .
